Question

guided notes lesson 8 - 3: aa similarity

i can prove triangle similarity using angle - angle. (ma.912.gr.1.2)

i can solve problems involving similarity in triangles. (ma.912.gr.1.6)

learn:
postulate 8.1: angle - angle (aa) similarity
if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
example: if $\angle a\cong\angle y$ and $\angle b\cong\angle z$, then $\triangle abc\cong\triangle yzx$.
when solving for unknown quantities in similar triangles, solving proportions will be utilized. see below for a review from algebra 1.
$\frac{x}{9}=\frac{2x - 3}{24}$
$24(x)=9(2x - 3)$ cross multiply
$24x = 18x-27$ distribute
$6x=-27$ subtract 18x from both sides
$x = - 4.5$ divide both sides by 6
practice solving proportions:

1. $\frac{t}{0.3}=\frac{17}{0.9}$
2. $\frac{x}{3}=\frac{3x + 2}{6}$
3. $\frac{5}{7k + 4}=\frac{2}{2k-3}$

example 1: determine whether the triangles are similar. explain your reasoning.
$\angle l\cong\angle l$ by the \underline{\qquad} property of congruence.
$\angle lpq\cong\underline{\qquad}$ by the corresponding angles theorem.
by aa similarity, \underline{\qquad}
example 1 check: determine whether the triangles are similar. explain your reasoning.
by the triangle angle - sum theorem, $m\angle a = \underline{\qquad}^\circ$ and $m\angle g=\underline{\qquad}^\circ$
because \underline{\qquad} two pairs of angles are congruent,
$\triangle abc$ is / is not similar to $\triangle fgd$. (circle the correct choice)

Explanation:

Practice Solving Proportions

Step1: Cross multiply to eliminate fractions

$\frac{t}{0.3} = \frac{17}{0.9} \\
0.9t = 0.3 \times 17$

Step2: Calculate right-hand side value

$0.9t = 5.1$

Step3: Isolate t by dividing by 0.9

$t = \frac{5.1}{0.9}$

Step4: Simplify the fraction

$t = \frac{17}{3} \approx 5.67$

---

Step1: Cross multiply to eliminate fractions

$\frac{x}{3} = \frac{3x+2}{6} \\
6x = 3(3x+2)$

Step2: Distribute the right-hand side

$6x = 9x + 6$

Step3: Isolate x terms

$6x - 9x = 6 \\
-3x = 6$

Step4: Solve for x

$x = \frac{6}{-3} = -2$

---

Step1: Cross multiply to eliminate fractions

$\frac{5}{7k+4} = \frac{2}{2k-3} \\
5(2k-3) = 2(7k+4)$

Step2: Distribute both sides

$10k - 15 = 14k + 8$

Step3: Isolate k terms

$10k - 14k = 8 + 15 \\
-4k = 23$

Step4: Solve for k

$k = -\frac{23}{4} = -5.75$

• The shared angle $\angle L$ is congruent to itself via the Reflexive Property of Congruence.
• Segments $PQ$ and $JK$ are parallel, so corresponding angles $\angle LPQ$ and $\angle LJK$ are congruent by the Corresponding Angles Theorem.
• With two pairs of congruent angles, the triangles satisfy AA Similarity.

Step1: Calculate $m\angle A$ using Triangle Angle-Sum Theorem

For $\triangle ABC$:
$m\angle A + 90^\circ + 44^\circ = 180^\circ \\
m\angle A = 180^\circ - 134^\circ = 46^\circ$

Step2: Calculate $m\angle G$ using Triangle Angle-Sum Theorem

For $\triangle FGD$:
$m\angle G + 90^\circ + 47^\circ = 180^\circ \\
m\angle G = 180^\circ - 137^\circ = 43^\circ$

Step3: Compare angle pairs

Only the right angles are congruent; the other angle pairs do not match, so there are not two pairs of congruent angles.

Answer:

1. $t = \frac{17}{3}$
2. $x = -2$
3. $k = -\frac{23}{4}$

---

Example 1